Autonomous systems are an emerging AI technology functioning without human intervention underpinned by the latest advances in intelligence, cognition, computer, and systems sciences. This paper explores the intelligent and mathematical foundations of autonomous systems. It focuses on structural and behavioral properties that constitute the intelligent power of autonomous systems. It explains how system intelligence aggregates from reflexive, imperative, adaptive intelligence to autonomous and cognitive intelligence. A hierarchical intelligence model (HIM) is introduced to elaborate the evolution of human and system intelligence as an inductive process. The properties of system autonomy are formally analyzed towards a wide range of applications in computational intelligence and systems engineering. Emerging paradigms of autonomous systems including brain-inspired systems, cognitive robots, and autonomous knowledge learning systems are described. Advances in autonomous systems will pave a way towards highly intelligent machines for augmenting human capabilities.